Restricted TS (BOLS)
Allocation probabilities of the TS are restricted to belong in the range \([\pi _{\text{min}}, \pi _{\text{max}}]\) , meaning that no allocation probabilities lower than \(\pi _{\text{min}}\)  or higher than \(\pi _{\text{max}}\)  are allowed. Formally, using the thresholds proposed in [1]}, TS's allocation probabilities \(\tilde{\pi }^{\text{TS}}_{t,k}\)  are redefined (clipped) as:
\(\pi _{\text{min}} &= 0.1; \quad \pi _{\text{max}} = 0.9\\\pi ^{\text{TS-BOLS}}_{t,k} &= \min \left(\pi _{\text{max}}, \max \left(\tilde{\pi }^{\text{TS}}_{t,k}, \pi _{\text{min}}\right)\right), \quad \forall t \in [0,T], k=0,1.\) 

In addition, in order to apply this batched estimation approach, it should be ensured that in each batch, each arm is allocated at least once, imposing an additional restriction on the TS algorithm.
Restricted TS (AW-AIPW)
Allocation probabilities of the TS are restricted to avoid a very fast decreasing or increasing rate towards the extremes \(0,1\) . Their redefinition is based on a time-varying function [2]}, that imposes the lower bound \((1/K) t^{-0.7}\) , with \(K\)  the number of arms, leading to:
\(\pi _{\text{min}} &= \frac{1}{K} t^{-0.7}; \quad \pi _{\text{max}} = 1- \pi _{\text{min}}\\\pi ^{\text{TS-AWAIPW}}_{t,k} &= \min \left(\pi _{\text{max}}, \max \left(\tilde{\pi }^{\text{TS}}_{t,k}, \pi _{\text{min}}\right)\right), \quad \forall t \in [0,T], k=0,1.\) 

